Proportion tutorial: the ol' single-exclamation proof of the Pythagorean Theorem.

"Behold!" is the traditional caption for this image, but that's just a snotty
mathemetician's way of saying "I worked out this algebra before you did!"

So: let the length of the sides of the whole image be A. The image is square.
Let the side length of the red square be B. The blue triangle is a right triangle.
It has sides of length A (on the outside), C (shortest), and D.

Cleverly, this diagram is drawn so that D = B + C. Any right triangle can be
have this diagram drawn about it. The Pythagorean theorem, for the blue triangle, is AxA = CxC + DxD.

The area of the big square is AxA. The area is also the area of the red square plus
4 of the blue triangles. The area of a triangle is .5x base x height. For a right triangle,
you can use either side touching the right angle as the height.
So: